Problems of Information Transmission

, Volume 54, Issue 2, pp 101–115 | Cite as

On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, q) and Extendability of Reed–Solomon Codes

  • D. BartoliEmail author
  • A. A. Davydov
  • S. Marcugini
  • F. Pambianco
Coding Theory


Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,
$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$
The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q−N+1]q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, qN + 2]q maximum distance separable code.


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  1. 1.
    Hirschfeld, J.W.P., Projective Geometries over Finite Fields, Oxford: Clarendon; New York: Oxford Univ. Press, 1998, 2nd ed.Google Scholar
  2. 2.
    Hirschfeld, J.W.P. and Storme, L., The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001, Finite Geometries (Proc. 4th Isle of Thorns Conf., July 16–21, 2000), Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., and Thas, J.A., Eds., Dev. Math., vol. 3, Dordrecht: Kluwer, 2001, pp. 201–246.Google Scholar
  3. 3.
    Hirschfeld, J.W.P. and Thas, J.A., Open Problems in Finite Projective Spaces, Finite Fields Appl., 2015, vol. 32, no. 1, pp. 44–81.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ball, S., Finite Geometry and Combinatorial Applications, Cambridge, UK: Cambridge Univ. Press, 2015.CrossRefzbMATHGoogle Scholar
  5. 5.
    Ball, S. and De Beule, J., On Subsets of the Normal Rational Curve, arXiv:1603.06714 [math.CO], 2016.zbMATHGoogle Scholar
  6. 6.
    Chowdhury, A., Inclusion Matrices and the MDS Conjecture, arXiv:1511.03623v3 [math.CO], 2015.zbMATHGoogle Scholar
  7. 7.
    Hirschfeld, J.W.P., Korchmáros, G., and Torres, F., Algebraic Curves over a Finite Field, Princeton: Princeton Univ. Press, 2008.CrossRefzbMATHGoogle Scholar
  8. 8.
    Klein, A. and Storme, L., Applications of Finite Geometry in Coding Theory and Cryptography, Information Security, Coding Theory and Related Combinatorics, Crnković, D. and Tonchev, V., Eds., NATO Sci. Peace Secur. Ser. D: Inf. Commun. Secur., vol. 29. Amsterdam: IOS Press, 2011, pp. 38–58.Google Scholar
  9. 9.
    Landjev, I. and Storme, L., Galois Geometry and Coding Theory, Current Research Topics in Galois Geometry, De Beule, J. and Storme, L., Eds., New York: Nova Science Pub., 2011, ch. 8, pp. 185–212.Google Scholar
  10. 10.
    MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.zbMATHGoogle Scholar
  11. 11.
    Roth, R.M., Introduction to Coding Theory, Cambridge: Cambridge Univ. Press, 2007.Google Scholar
  12. 12.
    Storme, L. and Thas, J.A., Complete k-Arcs in PG(n, q), q Even, Discrete Math., 1992, vol. 106/107, pp. 455–469.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Storme, L. and Thas, J.A., k-Arcs and Dual k-Arcs, Discrete Math., 1994, vol. 125, no. 1–3, pp. 357–370.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Thas, J.A., M.D.S. Codes and Arcs in Projective Spaces: A Survey, Matematiche (Catania), 1992, vol. 47, no. 2, pp. 315–328.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Segre, B., Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura Appl., 1955, vol. 39, no. 1, pp. 357–379.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ball, S., On Sets of Vectors of a Finite Vector Space in Which Every Subset of Basis Size is a Basis, J. Eur. Math. Soc., 2012, vol. 14, no. 3, pp. 733–748.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ball, S. and De Beule, J., On Sets of Vectors of a Finite Vector Space in Which Every Subset of Basis Size is a Basis. II, Des. Codes Cryptogr., 2012, vol. 65, no. 1, pp. 5–14.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Korchmáros, G., Storme, L., and Szőnyi, T., Space-Filling Subsets of a Normal Rational Curve, J. Statist. Plann. Inference, 1997, vol. 58, no. 1, pp. 93–110.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Storme, L., Completeness of Normal Rational Curves, J. Algebraic Combin., 1992, vol. 1, no. 2, pp. 197–202.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Storme, L. and Thas, J.A., Generalized Reed–Solomon Codes and Normal Rational Curves: An Improvement of Results by Seroussi and Roth, Advances in Finite Geometries and Designs (Proc. 3rd Isle of Thorns Conf., Chelwood Gate, UK, 1990), Hirschfeld, J.W.P., Hughes, D.R., and Thas, J.A., Eds., Oxford: Oxford Univ. Press, 1991, pp. 369–389.Google Scholar
  21. 21.
    Kovács, S.J., Small Saturated Sets in Finite Projective Planes, Rend. Mat. Appl., 1992, vol. 12, no. 1, pp. 157–164.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Bosma, W., Cannon, J., and Playoust, C., The Magma Algebra System, I: The User Language, J. Symbolic Comput., 1997, vol. 24, no. 3–4, pp. 235–265.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bartoli, D., Davydov, A.A., Faina, G., Kreshchuk, A.A., Marcugini, S., and Pambianco, F., Upper Bounds on the Smallest Size of a Complete Arc in a Finite Desarguesian Projective Plane Based on Computer Search, J. Geom., 2016, vol. 107, no. 1, pp. 89–117.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bartoli, D., Davydov, A.A., Marcugini, S., and Pambianco, F., On Almost Complete Subsets of a Conic in PG(2, q), Completeness of Normal Rational Curves and Extendability of Reed–Solomon Codes, arXiv:1609.05657v3 [math.CO], 2017.Google Scholar

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© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  • D. Bartoli
    • 1
    Email author
  • A. A. Davydov
    • 2
  • S. Marcugini
    • 1
  • F. Pambianco
    • 1
  1. 1.Department of Mathematics and Computer SciencesUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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